Rose-Hulman Undergraduate Mathematics Journal Volume 12 Issue 2 Article 6 The Edge Slide Graph of the 3-cube Lyndal Henden Massey University, Palmerston North, New Zealand, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Henden, Lyndal (2011) "The Edge Slide Graph of the 3-cube," Rose-Hulman Undergraduate Mathematics Journal: Vol. 12 : Iss. 2 , Article 6. Available at: https://scholar.rose-hulman.edu/rhumj/vol12/iss2/6 Rose- Hulman Undergraduate Mathematics Journal The Edge Slide Graph of the 3-cube Lyndal Hendena Volume 12, No. 2, Fall 2011 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Massey University, Palmerston North, New Zealand Rose-Hulman Undergraduate Mathematics Journal Volume 12, No. 2, Fall 2011 The Edge Slide Graph of the 3-cube Lyndal Henden Abstract. The goal of this paper is to study the spanning trees of the 3-cube by understanding their edge slide graph. A spanning tree of a graph G is a minimal set of edges that connects all vertices. An edge slide occurs in a spanning tree of the 3-cube when a single edge can be slid across a 2-dimensional face to form another spanning tree. The edge slide graph is the graph whose vertices are the spanning trees, with an edge between two vertices if the spanning trees are related by a single edge slide. This report completely determines the edge slide graph of the 3-cube. The edge slide graph of the 3-cube has twelve components isomorphic to the 4-cube, and three other components, mutually isomorphic, with 64 vertices each. The main result is to determine the structure of the three components that each have 64 vertices and we also describe their symmetries. Some partial results on the 4-cube are also provided. Acknowledgements: I would like to thank Dr. Christopher Tuffley for his help with this paper. 68 1 INTRODUCTION Contents 1 Introduction 68 2 Preliminaries 69 2.1 Edge slides and the edge slide graph . 69 2.2 Signatures . 71 2.3 Orientations and upright trees . 72 2.4 The edge slide graph of the 2-cube, Q2 ..................... 73 3 The edge slide graph of the 3-cube 73 3.1 Signatures (1; 2; 4) and (1; 3; 3) . 75 3.2 Signature (3,2,2) . 76 3.2.1 The Structure of E(3; 2; 2) . 81 4 Some partial results on the 4-cube. 83 1 Introduction The goal of this paper is to study the spanning trees of the 3-cube by understanding their edge slide graph. The 3-cube is the graph on the left in Figure 1, whose vertices and edges are the vertices and edges of an ordinary cube or die, and a spanning tree of a graph is a minimal set of edges that connects all of the vertices. In the case of the cube we may use the structure of the cube to define an edge slide operation transforming one tree into another: two trees are related by an edge slide if it is possible to transform one tree into the other by \sliding" an edge across a two-dimensional face. We may then look at the graph whose vertices are the spanning trees, with an edge between two trees if they are related by a single edge slide. The 3-cube is the third member of an infinite family of graphs fQng, where the n-cube Qn consists of the edges and vertices of an n-dimensional cube. The 3-cube Q3 and the 4-cube Q4 can be seen in Figure 1. The number of spanning trees of Qn may be found using Kirchhoff's Matrix Tree Theorem, and is known to be n 2n−n−1 Y (n) jTree(Qn)j = 2 k k (1) k=1 (see for example Stanley [3]). However, it is an open problem to find a bijective proof of the formula for the number of spanning trees of Qn. Tuffley [4] has recently found a combinatorial proof for counting spanning trees in the case where n = 3 using edge slides. There are difficulties when we try to generalise Tuffley's results for Q3 to higher di- mensions. Edge slides can be defined for higher dimensional cubes, however, some of the properties used to count the spanning trees of Q3 break down, and the argument cannot 69 1,2,3 1,2,3,4 1,2 1,2,3 8 < 8 < 8 < 8 < 2,3 1,2 2,3,4 8 < 1,2,4 8 < 8 < 8 < 1,3 1,3,4 2 8 < 8 < 1 1,3 8 < 2,3 2 2,4 8 < 8 < 8 < 8 < 8 < 1 3 3,4 1,4 8 < 8 < 8 < 8 < Æ Æ 3 4 8 < 8 < Figure 1: The 3-cube Q3 on the left, and the 4-cube Q4 on the right. readily be extended to higher dimensions. Our hope is that an understanding of the edge slide graph will give some insight into counting the spanning trees of Qn, for n ≥ 4. In this paper we completely determine the structure of the edge slide graph of Q3. We find that this has twelve components isomorphic to Q4, and three mutually isomorphic components that each have 64 vertices. In Section 2 we define terms and determine the structure of the edge slide graph of Q2, while in Section 3 we determine the structure of the edge slide graph of Q3. Finally, in Section 4 we present partial results on Q4. 2 Preliminaries 2.1 Edge slides and the edge slide graph n Definition 1. The n-cube is the graph Qn that has 2 vertices, the subsets of f1; 2; : : : ; ng. There is an edge between two vertices s and t if they differ by adding or removing one element. If s and t differ by adding or removing i, we will say that the edge fs; tg is in direction i. The 3-cube appears in Figure 1. Definition 2. A spanning tree of a graph G is a minimal set of edges that connects all vertices. See Figure 2 for an example. If G has v vertices then a spanning tree will have v − 1 edges (see for example Agnarsson and Greenlaw [1, p.98]). Definition 3. An edge slide occurs in a spanning tree T1 of Qn when a single edge can be slid across a two dimensional face to get a second spanning tree T2 in Qn. More precisely, an edge slide is the operation of switching between two edges of the form (s; s [ fig) and (s [ fjg; s [ fi; jg), where s ⊆ f1; 2; : : : ; ng, i; j 2 f1; 2; : : : ; ng, and i; j2 = s. An example is shown in Figure 3. 70 2 PRELIMINARIES 1,2 1,2,3 8 < 8 < 2 2,3 1 1,3 8 < 8 < 8 < 8 < 3 Æ 8 < Figure 2: The bold edges form a spanning tree of the 3-cube. 1,2 1,2,3 1,2 1,2,3 8 < 8 < 8 < 8 < 2 2 1 1,3 2,3 1 1,3 2,3 8 < 8 < 8 < 8 < 8 < 8 < 8 < 8 < Æ 3 Æ 3 8 < 8 < Figure 3: An edge slide from T1 to T2. This slide is an upward slide, because the edge moves from vertices f1,3g and f3g to vertices f1,2,3g and f2,3g, and the number of elements in each vertex has increased. 2.2 Signatures 71 1,2 1,2,3 1,2 1,2,3 8 < 8 < 8 < 8 < 2 2 1 1,3 2,3 1 1,3 2,3 8 < 8 < 8 < 8 < 8 < 8 < 8 < 8 < Æ 3 Æ 3 Figure 4: Spanning trees with8 different< signatures. The tree on the left has8 < one edge in direction 1, two edges in direction 2, and four edges in direction 3; its signature is (1,2,4). The tree on the right has three edges in direction 1 and two edges in each of directions 2 and 3; the signature is (3,2,2). A downward edge slide occurs when the number of elements in both vertices joined by the edge decreases when the edge is slid. In contrast, an upward edge slide occurs when the number of elements in each vertex increases when the edge is slid, as seen in Figure 3. We define the upper and lower faces of Qn with respect to direction i as the subgraphs induced by the vertices that respectively do and do not contain i. Then a downward edge slide in direction i moves an edge from the upper to the lower face with respect to direction i, while an upward edge slide moves an edge from the lower to the upper face. Definition 4. We define the edge slide graph of Qn, E(Qn), to be the graph whose vertices are the spanning trees of Qn, with an edge between T1 and T2 if they are related by a single edge slide. Thus, for example the edge slide graph of Q3 will have an edge between the vertices corresponding to the trees in Figure 3. 2.2 Signatures Definition 5. If a spanning tree T of Qn has ki edges in direction i then we define the signa- ture of T to be (k1; k2; : : : ; kn). Figure 4 shows two spanning trees with different signatures. Lemma 1 (Tuffley [4]). A spanning tree with ki edges in direction i will have at least ki − 1 edges that can be slid in direction i, or exactly ki − 1 if n = 2 or 3.